RESEARCH ARTICLESLight Propagation with Phasegles for total internal reflection,provided thatDiscontinuities:Generalized Laws ofθe=arcsin(3)Reflection and RefractionSimilarly,for reflection we havedΦ(4)where 0,is the angle of reflection.There is anonlinear relation betweenθ，amdO,which isConventional optical components rely on gradual phase shifts accumulated during lightmarkedly different from conventional specular re-propagation to shape light beams.New degrees of freedom are attained by introducing abruptflection.Equation 4 predicts that there is alwaysaphase changes over the scale of the wavelength.A two-dimensional array of optical resonatorscritical angle of incidencewith spatially varying phase response and subwavelength separation can imprint such phasediscontinuities on propagating light as it traverses the interface between two media.Anomalous(5)reflection and refraction phenomena are observed in this regime in optically thin arrays of metallicantennas on silicon with a linear phase variation along the interface,which are in excellentabove which the reflected beam becomesagreement with generalized laws derived from Fermat's principle.Phase discontinuities provideevanescent.great flexibility in the design of light beams,as illustrated by the generation of optical vorticesIn the above derivation.we have assumed thatthrough use of planar designer metallic interfaces.is a continuous function ofthe position alongthe interface;thus,all the incident energy is trans-he shaping of the wavefront of light withbetween two media;)depends on the co-ferred into the anomalous reflection and refraction.optical components such as lenses andordinate rs along the interface.Then,the totalHowever,because experimentally we use an arrayphase shift()+.dr will be stationaryof optically thin resonators with subwavelengthas gratings and holograms,relies on gradualphasefor the actual path that light takes is the waveseparation to achieve the phase change alongchanges accumulated along the optical path.Thisvector of the propagating light.This provides athe interface,this discreteness implies that thereownloadedapproach is generalized in transformation opticsgeneralization of the laws of reflection and re-are also regularly reflected and refracted beams,(1,2),which uses metamaterials to bend lightfraction,which is applicable to a wide range ofwhich follow conventional laws of reflectionin unusual ways,achieving such phenomena assubwavelength structured interfaces between twoand refraction (dd/dx=0 in Eqs.2 and 4).Thenegative refraction,subwavelength-focusing,andmedia throughout the optical spectrum.separation between the resonators controlscloaking (3,4)and even to explore unusual ge-Generalized laws of reflection and refraction.the amount of energy in the anomalously re-ometries of space-time in the early universe (5).The introduction of an abrupt phase shift,de-flected and refracted beams.We have alsoA new degree of freedom of controlling wave-noted as phase discontinuity,at the interface be-assumed that the amplitudes of the scatteredfronts can be attained by introducing abrupt phasetween two media allows us to revisit the laws ofradiation by each resonator are identical,so thatshifts over the scale of the wavelength along thereflection and refraction by applying Fermat'sthe reflected and refracted beams are plane waves.optical path,with the propagation of light gov-principle.Consider an incident plane wave at anIn the next section,we will show with simulations-emed by Fermat's principle.The latter states thatangle 0.Assuming that the two paths are infi-which represent numerical solutions of Maxwell'sthe trajectory taken between two points A and Bnitesimally close to the actual light path (Fig.1).by a ray of light is that of the least optical path,then the phase difference between them is zeroan()dr,where n)is the local index of re-fraction,and readily gives the laws of reflectionand refraction between two media.In its mostParis-[ko m sin(0 )dx+=0(1)general form,Fermat's principle can be stated asthe principle of stationary phase (6-8);that is,Saclayare,respectively,the phase discontinuities at thelated along the actual light path will be zero withlocations where the two paths cross the interface;ΦΦ+dΦrespect to infinitesimal variations of the path.Wedx is the distance between the crossing points;ninshow that an abrupt phase shift d(s)over theand n,are the refractive indices of the two media;scale of the wavelength can be introduced in theandk。=2πo,where入。is the vacuum wave-2022optical path by suitably engineering the interfacelength.If the phase gradient along the interface isdesigned to be constant,the previous equationleads to the generalized Snell's law of refractionsin(0)-sin(i)n=(2)Studies and Department of Physics Texas A&M University,BCollege Station,TX 77843,USA.Dipartimento di Fisica eEquation 2 implies that the refracted beam candelle Marde,via Brecce Bianche,60131 Ancona,Italy.Lab-have an arbitrary direction,provided that a suit-Fig.1.Schematics used to derive the generalizedable constant gradient ofphase discontinuity alongSnell's law of refraction.The interface between themale Superieure de Cachan and CNRS,94235 Cachan,France.Dipartimento di Fisica,Universita degli Studi di Trento,viathe interface (d/dx)is introduced.Because oftwo media is artificially structured in order to in-Sommarive 14,38100 Trento,Italy.the nonzero phase gradient in this modified Snell'stroduce an abrupt phase shift in the light path,"To whom correspondence should be addressed.E-maillaw,the two angles of incidence t0;lead to dif-which is a function of the position along the in-ferent values for the angle of refraction.As a terface.d and d+d are the phase shifts whereedu (Z.G.)consequence,there are two possible critical an-the two paths (blue and red)cross the boundary.www.sciencemag.orgSCIENCE VOL 334 21 OCTOBER 2011333RESEARCH ARTICLESequations-how,indeed,one can achieve theacross aresonance.By spatiallytailoring the geom-Phase shifts covering the 0-to-2 range areequal-amplitude condition and the constant phaseetry of the resonators in an array and hence theirgradient along the interface through suitable de-frequency response,one can design the phaseTo achieve the required phase coverage whilesign of the resonators.discontinuity along the interface and mold themaintaining large scattering amplitudes,we usedThere is a fundamental difference between thewavefront of the reflected and refracted beams inthe double-resonance properties of V-shaped an-anomalous refraction phenomena caused by phasenearly arbitrary ways.The choice of the reso-tennas,which consist of two arms ofequal lengthdiscontinuities and those found in bulk designernators is potentially wide-ranging,from electro-h connected at one end at an angle A(Fig.2B).metamaterials,which are caused by either negativemagnetic cavities (9,10)to nanoparticle clustersWe define two unit vectors to describe the ori-dielectric permittivity and negative magnetic(1/,12)and plasmonic antennas (13,14).Weentation of a V-antenna:$alongthe symmetry axispemeability or anisotropic dielectric pemittivityconcentrated on the latter because of their widelyof the antenna and a perpendicular tos(Fig.2B).with different signs of permittivity tensor com-tailorable optical properties(15-19)and the easeV-antennas support“symmetric'”and"“antisym-ponents along and transverse to the surface (3,4).of fabricating planar antennas of nanoscale thick-metric"modes(Fig.2B,middle and right),whichPhase response of optical antennas.The phaseness.The resonant nature of a rod antenna madeare excited by electric-field components alongsshift between the emitted and the incident radia-of a perfect electric conductor is shown in Fig.and a axes,respectively.In the symmetric mode,tion of an optical resonator changes appreciably2A(20.the current distribution in each arm approximatesFig.2.(A)Calculated phase and amplitude ofSymmetric modeAntisymmetric modeAscattered light from a straight rod antenna made ofa perfect electric conductor (20).The vertical dashedline indicates the first-order dipolar resonance of1.0-phasethe antenna.(B)A V-antenna supports symmetricamplitudeand antisymmetric modes,which are excited,re-spectively,by components of the incdent field alongs and a axes.The angle between the incident po-larization and the antenna symmetry axis is 45.The schematic current distribution is representedby colors on the antenna (blue for symmetric and0.040.20.60.81.0red for antisymmetric mode),with brighter color0.4representing larger currents.The direction of cur-rent flow is indicated by arrows with color gradient.DAmplitude EPhase shift(O V-antennas corresponding to mirror images of(normalized）those in (B).The components of the scattered elec-1.871.8https://tric field perpendicular to the inddent field in (B)0.916135and (C)have a phase difference.(D and E)An-0.8alytically calculated amplitude and phase shift of0.71490the cross-polarized scattered light for V-antennas0.60.5and with various length h and angle between the0.412451.00rods△ato=8μm(20).The four circles in(D)and0.80.30.845(日indicate the values of h and△used in exper-iments.The rod geometry enables analytical cal-0.6-0.6-900.1Univculations of the phase and amplitude of the scattered0.400.4+135light,without requiring the extensive numerical020406080100120140160180020406080100120140160180simulations needed to compute the same quan-A(degree)△(degree)Paristities for "flat"antennas with a rectangular cross-section,as used in the experiments.The opticalproperties of a rod and "flat"antenna of the samelength are quantitatively very similar,when therod antenna diameter and the "flat"antenna40width and thickness are much smaller than thelength (20).(F)Schematic unit cell of the plasmonicinterface for demonstrating the generalized laws ofreflection and refraction.The sample shown in Fig.3A2022the unit cell.The antennas are designed to haveequal scattering amplitudes and constant phaseN10difference△Φ=π/4 between neighbors.(G)Finite-difference time-domain (FDTD)simulations of thescattered electric field for the individual antennascomposing the array in (F).Plots show the scat-tered electric field polarized in the x direction fory-polarized plane wave excitation at normal in-cidence from the silicon substrate.The siliconsubstrate is located at z s 0.The antennas are equally spaced at a sub-principle,the anomalously refracted beam resulting from the superposi-wavelength separation 1/8,where I is the unit cell length.The tilted redtion of these spherical waves is then a plane wave that satisfies thestraight line in(G)is the envelope of the projections of the spherical wavesgeneralized Snell's law (Eq.2)with a phase gradient ldddxl=2/alongscattered by the antennas onto the x-z plane.On account of Huygens'sthe interface.33421 OCTOBER 2011 VOL 334 SCIENCEwww.sciencemag.orgRESEARCH ARTICLESthat of an individual straight antenna of lengthlytical calculations of the amplitude and phaseExperiments on anomalous reflection andh(Fig.2B,middle),and therefore the first-orderresponse of V-antennas assumed to be made ofantenna resonance occurs at h≈，e/2,wheregold rods are shown in Fig.2,D and E.In Fig.generalized laws of reflection and refractionAe is the effective wavelength (/4).In the anti-2D,the blue and red dashed curves correspond tousing plasmonic interfaces constructed by peri-symmetric mode,the current distribution ineachodically arranging the eight constituent antennasarm approximates that of one half of a straightsymmetric modes,respectively.We chose fouras explained in the caption of Fig.2F.The spacingantenna of length 2h(Fig.2B,right),and theantennas detuned from the resonance peaks,asbetween the antennas should be subwavelengthcondition for the first-order resonance of thisindicated by circles in Fig.2,D and E.whichso as to provide efficient scattering and to preventmode is2h≈e2provide an incremental phase of n4 from left tothe occurrence of grating diffraction.However,itThe polarization of the scattered radiationright for the cross-polarized scattered light.Byshould not be too small;otherwise,the strong near-is the same as that of the incident light whensimply taking the mirror structure(Fig.2C)of anfield coupling between neighboring antennasthe latter is polarized along s or a.For an ar-existing V-antenna (Fig.2B),one creates a newwould perturb the designed scattering amplitudesbitrary incident polarization,both antenna modesantenna whose cross-polarized radiation has an ad-and phases.A representative sample with theare excited but with substantially different am-ditional n phase shift.This is evident by observingdensest packing of antennas,I=11 um,is shownplitude and phase because of their distinctive reso-that the currents leading to cross-polarized radia-in Fig.3A,where I is the lateral period of thenance conditions.As a result,the scattered lighttion are t out ofphase in Fig.2,B and C.A set ofantenna array.In the schematic of the experimen-can have a polarization different from that of theeight antennas were thus created from the initialtal setup (Fig.3B),we assume that the cross-incident light.These modal properties of thefour antennas,as shown in Fig.2F.Full-wave sim-polarized scattered light from the antennas on theV-antennas allow one to design the amplitude.ulations confim that the amplitudes of the cross-left side is phase-delayed as compared with thephase,and polarization state ofthe scattered light.polarized radiation scattered by the eight antennas areones on the right.By substituting into Eq.2-2m/TWe chose the incident polarization to be at 45nearly equal,with phases in t/4 increments (Fig.2G).for dd/dx and the refractive indices of siliconwith respect to and a so that both the symmetricA large phase coverage (~300)can also beand air (nsi and 1)for n and n,we obtain theand antisymmetric modes can be excited andachieved by using arrays of straight antennas (fig.angle ofrefraction for the cross-polarized beamthe scattered light has a substantial componentS3).However,to obtain the same range of phasepolarized orthogonal to that of the incident light.shift their scattering amplitudes will be substanExperimentally,this allows us to use a polarizertially smaller than those of V-antennas (fig.S3).to decouple the scattered light from the excitation.As a consequence of its double resonances,theFigure 3C summarizes the experimental resultsAs a result of the modal properties of theV-antenna instead allows one to design an arrayof the ordinary and the anomalous refraction forwnloadedV-antennas and the degrees of freedom in choosingwith phase coverage of 2nt and equal,yet high,six samples with different I at normal incidence.antenna geometry(hand△，the cross-polarizedscattering amplitudes for all of the amay elements.The incident polarization is along the y axis inscattered light can have a large range of phasesleading to anomalously reflected and refractedFig.3A.The sample with the smallest I corre-and amplitudes for a given wavelength Ao;ana-beams of substantially higher intensities.sponds to the largest phase gradient and the mostFig.3.(A)Scanning electron microscope (SEM)Bimage of a representative antenna array fabricatedy-polarized incidence fromon a silicon wafer.The unit cell of the plasmoniccollimated quantuminterface (yellow)comprises eight gold V-antennascascade laser(.=8μmw.science.orgof width-220 nm and thickness-50 nm,and itordinaryrepeats with a periodicity of I=11 um in the xanomalousreflectiondirection and 1.5 um in they direction.(B)Schematicreflectionfield normal to the plane of incidence).(C and D)gradient ofMeasured far-field intensity profiles of the refractedphase shiftbeams for y-and x-polarized excitations,respective-dΦ<0ly.The refraction angle is counted from the normaldxto the surface.The red and black curves are mea-sured with and without a polarizer,respectively,foranomaloussix samples with different I.The polarizer is used torefractionordinaryselect the anomalously refracted beams that arerefractioncross-polarized with respedt to the excitation.TheApril 21,amplitude of the red curves is magnified by a factorof two for clarity.The gray arrows indicate the0.=0,y-polarized excitationD0,=0,x-polarized excitation2022calculated angles of anomalous refraction accordingr (um)r (um)to Eq.6.1111る013401515401717451919210221-60-40-200204060-6040-200204060Refraction angle (degree)Refraction angle (degree)www.sciencemag.orgSCIENCE VOL 334 21 OCTOBER 2011335